Abstract

This paper concerns an NN-order problem in the calculus of variations of minimizing the functional ∫abΛ(t,x(t),…,x(N)(t))dt\smash{\int_{a}^{b}{\Lambda(t,x(t),\ldots,x^{(N)}(t))\mathrm{d}t}}, in which the Lagrangian Λ\Lambda is a Borel measurable, non autonomous, and possibly extended valued function. Imposing some additional assumptions on the Lagrangian, such as an integrable boundedness of the partial proximal subgradients (up to the (N ⁣− ⁣2N\!-\!2)-order variable), a growth condition (more general than superlinearity w.r.t. the last variable) and, when the Lagrangian is extended valued, the lower semicontinuity, we prove that the NN-th derivative of a reference minimizer is essentially bounded. We also provide necessary optimality conditions in the Euler-Lagrange form and, for the first time for higher order problems, in the Erdmann-Du Bois-Reymond form. The latter can be also expressed in terms of a (generalized) convex subdifferential, and is valid even without requiring neither a particular growth condition nor convexity in any variable.

Author information

Contact details are reproduced from the original publication and may be historical.

Julien Bernis

Laboratoire de Mathématiques, Université de Brest, 29200 Brest, France
and: UMR CNRS 6205, Brest, France

julien.bernis@univ-brest.fr

Keywords

  • Calculus of variations
  • minimizer regularity
  • higher order problems
  • necessary conditions
  • Weierstrass inequality
  • Erdmann-Du Bois-Reymond condition

Mathematics Subject Classification

49N60, 49K15

References

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Suggested citation

J. Bernis, P. Bettiol, C. Mariconda. “Higher Order Problems in the Calculus of Variations: Du Bois-Reymond Condition and Regularity of Minimizers.” Journal of Convex Analysis 27 (2020), No. 1, 179–204. https://doi.org/10.68381/jca27011

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press.